2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, 
BC, Canada.
Electronic Journal of Differential Equations,
Conference 12, 2005, pp. 171-180. 

Title: Oscillation criteria for functional differential equations

Author: Ioannis P. Stavroulakis (Univ. of Ioannina, Greece)
 
Abstract: 
Consider the first-order linear delay differential equation 
$$
x'(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0},
$$
and the second-order linear delay equation 
$$
x''(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0},
$$
where $p$ and $\tau $ are continuous functions on $[t_{0},\infty )$, 
 $p(t)>0$, $\tau (t)$ is non-decreasing, $\tau (t)\leq t$ for $t\geq t_{0}$
and $\lim_{t\to \infty }\tau (t)=\infty $. Several oscillation
criteria are presented for the first-order equation when 
$$
0<\liminf_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds\leq \frac{1}{e}
\quad \hbox{and}\quad \limsup_{t\to \infty }\int_{\tau
(t)}^{t}p(s)ds<1,
$$
and for the second-order equation when 
$$
\liminf_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds
\leq \frac{1}{e}\quad \hbox{and}\quad \limsup_{t\to \infty }\int_{\tau
(t)}^{t}\tau (s)p(s)ds<1\,.
$$

Published April 20, 2005.
Math Subject Classifications: 34K11, 34C10.
Key Words: Oscillation; delay differential equations.